Inflation in an open universe

Motivated by observational indications of low cosmological mass density, we study a spatially open inflation modified hot big bang model whose evolutionary history is divided into three epochs: an early scalar field inflation epoch and the usual radiation and baryon (nonrelativistic matter) epochs. Generalizing techniques previously developed, we derive general solutions of the relativistic linear peturbation equations in each epoch. The constants of integration in the inflation epoch solutions are determined from quantum-mechanical initial conditions under the assumption that the perturbations are in the ground state at early times. The constants of integration in the radiation and baryon epoch solutions are determined from joining conditions derived by requiring that the linear perturbation equations remain nonsingular at the transitions between epochs. Expressions are derived for a number of baryon-epoch statistics which characterize large-scale structure, including the fractional mass, peculiar velocity, and gravitational potential perturbation two-point correlation functions, and the mean square value of the fractional mass and peculiar velocity perturbations. The Sachs-Wolfe relation is generalized to the open model and an expression for the angular fluctuation spectrum of the cosmic microwave background radiation temperature anisotropy is derived; we also determine the dipole velocity perturbation two-point correlation function and mean square value.The fractional energy density perturbation power spectrum is not a power law; on small scales we find the usual n=+1 scale-invariant flat model law, while on large scales we discover a n=-1 spectrum. The small-scale part of the fractional mass perturbation two-point correlation function agrees with what one finds in the n=+1 scale-invariant flat case, but it has a second zero and at large scales the fractional mass perturbations are weakly positively correlated. Once again, given the form of the fractional mass perturbations are weakly positively correlated. Once again, given the form of the fractional energy density perturbation power spectrum in this model, we find that the slope of the inflation epoch scalar field potential cannot be unduly constrained by observational data.